nasirafaf.substack.com/p/the-k2-abruzzi-route-as-analogy-for
3 corrections found
Their key innovation was proving that a carefully tailored, smooth external forcing term could drive a fluid into a finite-time singularity (blow-up) without violating physical constraints.
This misattributes the smooth-forcing breakthrough. Buckmaster says Córdoba and Martínez-Zoroa’s program achieved blow-up with rough forcing, while Buckmaster and Alpöge pushed that program to smooth forcing and to the Euler equations.
Full reasoning
Buckmaster’s own September 2026 statement draws a clear distinction between the earlier Córdoba–Martínez-Zoroa program and the later Buckmaster–Alpöge extension. He writes that Córdoba and Martínez-Zoroa’s program had already achieved blow-up results with rough forcing, and that he and Alpöge, with LLM assistance, pushed it to smooth forcing and to the incompressible Euler equations.
That matches Tao’s earlier summary of the prior Córdoba–Martínez-Zoroa result: Tao described their forced-Euler construction as keeping the forcing term bounded in C^{1,α} for α < 1/2, which is a regularity condition weaker than smoothness. So the article’s sentence incorrectly credits Córdoba and Martínez-Zoroa with the specific smooth-forcing breakthrough that Buckmaster attributes to the later Buckmaster–Alpöge work.
2 sources
- Statement by Tristan Buckmaster
“what Levent and I did was to take the Córdoba and Martínez-Zoroa program, which achieved blowup results with rough forcing, and ... push it to smooth forcing and to the incompressible Euler equations.”
- Terence Tao, 'Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations'
Tao summarizes the earlier Córdoba–Martínez-Zoroa result as one where “the forcing term f stays uniformly bounded in C^{1,α}(R^3) for any α < 1/2,” i.e. not a smooth-forcing result.
At that point, it appears that OpenAI jumped in and released a swarm of 10,000 artificial AI agents who managed to crack the problem, thereby laying credit at OpenAI's door.
This places OpenAI’s swarm effort at the time of the August 15 breakthrough, but OpenAI says the Navier–Stokes push began on September 1 and reached a result on September 5.
Full reasoning
The timeline in the post is off. The sentence follows immediately after the August 15 Buckmaster–Alpöge breakthrough and says “At that point” OpenAI launched the 10,000-agent effort. But OpenAI’s own chronology says otherwise: it heard rumors on September 1, 2026, launched the Millennium-problem evaluation then, and the Navier–Stokes agents reached their resolution on September 5, 2026.
Buckmaster’s statement is consistent with that later start date. He says he first wrote to OpenAI on September 3, spoke with them on September 6, and was eventually told their first prompt had been sent only “in the past few days,” after information about Buckmaster and Alpöge’s work had reached OpenAI. So there is concrete evidence that the OpenAI swarm did not begin back on August 15.
2 sources
- OpenAI, 'On the Navier–Stokes Millennium Prize Problem'
“On Tuesday, September 1, we heard rumors ... we launched an effort...” and “The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched.”
- Statement by Tristan Buckmaster
Buckmaster says he wrote to OpenAI on “Thursday, September 3rd,” spoke with them on “Sunday, September 6th,” and that OpenAI eventually said its first prompt had been sent only “in the past few days.”
OpenAI spins up an 88-hour swarm run consuming roughly 130 billion tokens to tackle and formalize the viscous Navier–Stokes problem in Lean.
OpenAI did not say the entire Lean formalization was part of the 88-hour run. It said the Navier–Stokes resolution took about 88 hours, and Lean formalization and verification took an additional 17 hours.
Full reasoning
OpenAI’s published timeline separates the discovery phase from the Lean phase. According to the company, the agents reached their Navier–Stokes resolution after about 88 hours. Only after that did Lean formalization and verification happen, taking an additional 17 hours via GPT‑6 Astra.
OpenAI also says the approximate 130 billion output tokens were used “in the process of resolving the Navier–Stokes problem.” So the article compresses two distinct stages into one and incorrectly implies the full Lean formalization was completed inside the 88-hour swarm run.
2 sources
- OpenAI, 'On the Navier–Stokes Millennium Prize Problem'
“The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT‑6 Astra.”
- OpenAI, 'On the Navier–Stokes Millennium Prize Problem'
“In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens.”